Lie Ideals and Generalized Derivations in Semiprime Rings

Notes on Generalized Derivations on Lie Ideals in Prime Rings

Let R be a prime ring, H a generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that usH(u)ut = 0 for all u ∈ L, where s ≥ 0, t ≥ 0 are fixed integers. Then H(x) = 0 for all x ∈ R unless char R = 2 and R satisfies S4, the standard identity in four variables. Let R be an associative ring with center Z(R). For x, y ∈ R, the commutator xy− yx will be denoted by [x, y]. An add...

Generalized Derivations in Semiprime Gamma Rings

On Generalized Derivations of Semiprime Rings

Let F be a commuting generalized derivation, with associated derivation d, on a semiprime ring R. We show that d(x)[y, z] = 0 for all x, y, z ∈ R and d is central. We define and characterize dependent elements of F and investigate a decomposition of R relative to F . Mathematics Subject Classification: 16N60, 16W25

Remarks on Generalized Derivations in Prime and Semiprime Rings

Let R be a ring with center Z and I a nonzero ideal of R. An additive mapping F : R → R is called a generalized derivation of R if there exists a derivation d : R → R such that F xy F x y xd y for all x, y ∈ R. In the present paper, we prove that if F x, y ± x, y for all x, y ∈ I or F x ◦ y ± x ◦ y for all x, y ∈ I, then the semiprime ring R must contains a nonzero central ideal, provided d I /...

Derivations in semiprime rings and Banach algebras

Let $R$ be a 2-torsion free semiprime ring with extended centroid $C$, $U$ the Utumi quotient ring of $R$ and $m,n>0$ are fixed integers. We show that if $R$ admits derivation $d$ such that $b[[d(x), x]_n,[y,d(y)]_m]=0$ for all $x,yin R$ where $0neq bin R$, then there exists a central idempotent element $e$ of $U$ such that $eU$ is commutative ring and $d$ induce a zero derivation on $(1-e)U$. ...

On Jordan generalized k-derivations of semiprime Γ_N-Rings